On the residual Monge-Ampère mass of plurisubharmonic functions, III: uniformly directional Lipschitz

The purpose of this article is to study the (residual) Monge-Ampère mass of a plurisubharmonic function with an isolated unbounded locus. A general decomposition formula is obtained under the Sasakian structure of the unit sphere. In complex dimension two, we obtain an $L^1$-apriori estimate on the complex Monge-Ampère operator. This induces an upper-bound estimate on the residual mass, provided with the uniform directional Lipschitz continuity. As an application, the zero mass conjecture is confirmed, if the function further separates the circular direction in its alternating part.